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# Doc5 - 2 Find the unique solution for each of the following...

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Unformatted text preview: 2. Find the unique solution for each of the following recur- rence relations. a} on“ — 1.5::n = ﬂ, rt 30 lad-41.1,, — 5d,,_. = l3, rt 3] e] 3a,,“ #441,, =0, nail, n; =5 (1} En” —3o,,_. =0, :23 i, :14 =3] 4. The number of bacteria in a culture is 1000 {approximately}, and this number increases 259% ever},f two hours. Use a recur- rence relation to determine the number of bacteria present after one day. 10. Fora 2: Lapermutation pl, p2, p3. , , . , p" oftheintegers, 2, 3, . , , , n is called orderly if, for each i = l, 2, 3, . . ., n — Lthere esistsaj o» isUchthatIpJ. — p,| = i.[lf.n = 2,1]1e permutations 1, land 2, l are both orderly. Whenn = 3we ﬁnd that 3, l, 2 is an orderly permutation, while 2, 3, 1 is not. [‘d‘r‘h}r not?)] [a] List all the orllzlerlg,r permutations for i, 2, 3. {In} List all the orderly permutations for l, 2. 3, 4. {c} If p], p2, p3, p4, p5 is an orderlyr permutation of l, 2, 3, 4, 5, what value[s] can pl be? {d} For rt :- 1, let an count the number of orderlyr permu- tations for l, 2, 3, .. . , n. Find and solve a recurrence relation for on. 16. For n 3 i. let on be the number of ways to write a as an or- dered sum of positive integers, where each summand is at least 2. (For example. :15 = 3 becaUse here we may represent 5 by 5, by 2 + 3, and by 3 + 2.) Find and solve a remrrence relation for an. 24. Porn 3 l, let can count the number of ways to tile 3 2 X n chessboard using horizontal (1 X 2] dominoes [which can also be used as vertical (2 X l] dominoes] and square [2 X 2) tiles. Find and solve a recurrence relation for a". 26. Let}: = {f}. l} andrt = [[1, [i], ii} g E*.Forn 31,]etnn count the number of shings in A“ of length it. Find and solve a recurrence relation for on . 10. The general solution of the recurrence relation a”: + blots“ +b2an = bgn + .54, n 30, with b. constant for l 5 i 5 4, is (3.2” +1323" +n — 7. Find .6, for each I ﬁt 54. 12. Let E = {[1, L 2, 3}. Fern 3 Lletnr,1 count the number of strings in E" containing an odd number of 1’s. Find and solve a recurrence relation for an. ...
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